2021 April MIMC 10 Problems/Problem 11
How many factors of is a perfect cube or a perfect square?
Solution
We need to calculate the number of perfect squares and the number of perfect cubes and then subtract the number of th power according to the principle of inclusion and exclusion. First of all, we need to factor
. Since we can choose even amount of each factor, there are a total of
perfect squares. Using the same logic, any number that is a cube must have multiple of
factors for each factor. Therefore, there are
cubes. In addition, there are
numbers with
th power. In total, there are
perfect square or perfect cube factors of
.